Start with:
Nearby point:
Average slope:
Start with a point \((x,f(x))\) on the graph.
A nearby point can be \((x+\Delta x, f(x+\Delta x))\), where \(\Delta x\) is a short distance from \(x\).
The average slope between \((x,f(x))\) and \((x+\Delta x, f(x+\Delta x))\) is given by
\[\text{average slope} = \frac{f(x+\Delta x) - f(x)}{(x+\Delta x)-x} = \frac{f(x+\Delta x) - f(x)}{\Delta x}.\]
What happens when \(\Delta x\) becomes smaller and smaller?